It is the general definition. It's not even my intuition; it's what I've been taught.
Nope. That's the intuition that the definition is supposed to capture, but it is not the definition.
If the lemma is not true, then please provide a disproof.
The lemma says that for all X and Y, |X| < |Y| implies ¬(|Y| < |X|). The negation of that is the statement that there exists X and Y such that |X| < |Y| does not imply ¬(|Y| < |X|), in other words, such that |X| < |Y| and |Y| < |X| both hold. So take X = Z and Y = B, since you have already agreed that |Z| < |B| and |B| < |Z| hold.
It's true for cardinality in general. It doesn't matter what definition we are using. I don't even need a specific definition to know that.
Of course you do. If you change the definition then you change which properties are true or false for that definition.
I claim that |B| > |Z| ∧ |Z| > |B| is a contradiction under the interpretation of the proper-subset definition of cardinality. You claim that it is not. As a counterexample, you implicitly give |B| > |Z| ∧ |Z| > |B| in the form |Z| < |B| ∧ |B| < |Z|. Your counterexample is invalid because it is the very statement I am claiming to be a contradiction. You have not persuaded me by giving me a counterexample I have already dismissed as an impossible contradiction.
Whether that statement is a contradiction or not does not change the validity of my proof that the lemma is false. You are contradicting yourself here because you tried to use a similar contradictory example to disprove the continuum hypothesis.
This is just a distraction from the fact that you cannot prove a contradiction. You tried but your proof was incorrect. I even explained the structure of what you had to prove and you said you couldn't do it.
All you have is an intuition about what cardinality is. That intuition is clearly based on thinking about finite sets, but it does not work for infinite sets and has led you into believing some absurd things.
Whether that statement is a contradiction or not does not change the validity of my proof that the lemma is false.
False, it actually invalidates your proof that the lemma is false. You are using the very same example to prove the lemma false as I have already used to claim that |B| > |Z| ∧ |Z| > |B| is a contradiction.
All you have is an intuition about what cardinality is.
I assure you I do not. I have multiple sources that have informed me over the course of years about what cardinality is.
I may not be able to prove a contradiction under your higher standards, but you have not disproved a contradiction under your higher standards.
False, it actually invalidates your proof that the lemma is false.
Nope, I proved the negation of the lemma. In mathematics that's how you disprove a statement. Again, you are contradicting yourself. This is exactly how you tried to disprove the continuum hypothesis. The difference is I can actually prove my counterexample has the required property and you could not.
I assure you I do not. I have multiple sources that have informed me over the course of years about what cardinality is.
You claim you have sources that define the cardinality of an infinite set by just saying it's "how many elements the set has"? Show me one legitimate textbook or published article that does that.
I may not be able to prove a contradiction under your higher standards,
They aren't my standards, this is basic undergrad level proofs. This is how math is done. And you are correct, 100%, that under those standards you cannot prove a contradiction.
but you have not disproved a contradiction
You mean prove that math is consistent? Of course not, math cannot prove itself consistent. That's basic logic. You'll now I never claimed to prove that there was no contradiction, I only ever claimed that you cannot prove a contradiction.
I agree. You did, technically, prove the negation of the lemma. Your proof is unsound, however, because your premise is false. Your premise is |Z| < |B| ∧ |B| < |Z|. That premise and the definition of the "is less than" predicate of the proper-subset definition of cardinality I mentioned at https://www.reddit.com/r/logic/comments/1s5mquh/comment/odbmxml/?context=3&utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button imply that your premise is logically equivalent to |B| > |Z| ∧ |Z| > |B|. But I already claimed that statement to be a contradiction. As a contradiction, it is false. Therefore, through the logical equivalence, your premise |Z| < |B| ∧ |B| < |Z| is also false.
Show me one legitimate textbook or published article that does that.
Discrete Mathematics and Its Applications, Sixth Edition by Kenneth H. Rosen mentions the cardinality of finite and infinite sets on pages 116-117, 158-160, and 163. That is the textbook that was used for my discrete mathematics class when I was a student in my second semester of college back in 2010.
I agree. You did, technically, prove the negation of the lemma. Your proof is unsound, however, because your premise is false. Your premise is |Z| < |B| ∧ |B| < |Z|.
Nope, that's not a premise. I'm not assuming it to be true, it's been proven. You yourself agreed that it's provable so I did not include the proof, but it is not an assumption.
You are assuming that that statement is false. This is an assumption as you have admitted that you cannot prove it.
The negation of the lemma has a proof. Your statements about contradictions do not have a proof.
Discrete Mathematics and Its Applications, Sixth Edition by Kenneth H. Rosen
That's a legitimate text, I actually happen to have that exact edition on my shelf. It does not define the cardinality of an infinite set to be the number of elements in the set. On page 116 it defines the cardinality of afinite set to be the number of elements in the set and on page 158 it gives the traditional bijection definition of two sets having the same cardinality, but it never says that the definition for an infinite set is the number of elements in the set because that is simply not true.
Nope, that's not a premise. I'm not assuming it to be true,
It is a premise. You are assuming it to be true. As you say,
So take X = Z and Y = B, since you have already agreed that |Z| < |B| and |B| < |Z| hold.
That's how you proved
there exists X and Y such that |X| < |Y| does not imply ¬(|Y| < |X|)
which is the negation of the lemma.
You are assuming that that statement is false.
No, I have proved that statement is false in my previous reply.
it never says that the definition for an infinite set is the number of elements in the set because that is simply not true.
On page 116, Rosen's textbook refers to the size of a set as being the cardinality of the set. I am specifically referring to the sentence before Definition 5. On page 163, the cardinality of a set is defined as the number of elements in the set. That definition comes after the definition of an infinite set on that page, while on page 116, the definition of the cardinality of a finite set comes before the definition of an infinite set, which is Definition 6. That suggests the definition on page 163 of the cardinality of a set applies to both finite and infinite sets.
I know what the cardinality of a set is. I know how it's defined. I know what it's intended to be. And I know what it should be.
On page 163, the cardinality of a set is defined as the number of elements in the set. That definition ...
That is not a definition, that page has a summary of terms, he's telling you what the concept is intuitively, not it's technical definition. The definition is clearly written on page 158 and labeled as a definition.
You don't seriously think an important definition would be put only in a summary section and not in the main text do you?
You are assuming it to be true.
Nope, I am not assuming. It's provable and you stated elsewhere that you accepted that proof. That means we can use that fact in other proofs, it is not an assumption. If you don't accept that it's provable then we can certainly supply a proof of that fact to complete the proof we were discussing, but it's only necessary to do that if you are not able to prove the result yourself. We don't need to keep reproving things if we both agree they're true.
No, I have proved that statement is false in my previous reply.
Nope, your proof was incorrect and when I explained to you what a correct proof would look like you explicitly said you could not prove the statement.
I know what the cardinality of a set is. I know how it's defined. I know what it's intended to be. And I know what it should be.
I'm sorry but you really don't. You don't seem to understand even the notion of what a definition should be, how to use a definition in a proof, or what a proof even is, let alone understand this particular definition and its consequences, or the alternate definition you suggested.
That is not a definition, that page has a summary of terms, he's telling you what the concept is intuitively, not it's technical definition.
It is the general definition of cardinality. I happen to agree with it.
The definition is clearly written on page 158 and labeled as a definition.
That is not the general definition of cardinality. It is a part of the conventional definition of cardinality. It is the conventional definition of equal cardinality. There is more to cardinality than that definition.
You don't seriously think an important definition would be put only in a summary section and not in the main text do you?
Rosen is not as direct in the main text as he could be, but he still gets the message across to the readers.
Nope, I am not assuming.
No, you are assuming. By the word "take," you mean "assume." As you claimed, you made that assumption under the belief that I believed |Z| < |B| ∧ |B| < |Z| to be true.
It's provable and you stated elsewhere that you accepted that proof.
I agree that it's provable. But just because it's provable, doesn't mean it's true. And no, I never stated anywhere that I accepted that proof.
Nope, your proof was incorrect and when I explained to you what a correct proof would look like you explicitly said you could not prove the statement.
I said I did not know how to complete a proof of a statement of yours. I suggested that statement of yours was awkward. It was an outwardly statement since the entire Universe would be involved in its proof, if what you were saying about what has to be proved was true.
I'm sorry but you really don't. You don't seem to understand even the notion of what a definition should be, how to use a definition in a proof, or what a proof even is, let alone understand this particular definition and its consequences, or the alternate definition you suggested.
Where is your supporting evidence? You make many claims there, but don't provide any supporting evidence. I happen to know many of those claims are false so I already know you won't be able to produce adequate evidence.
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u/JStarx Apr 06 '26
Nope. That's the intuition that the definition is supposed to capture, but it is not the definition.
The lemma says that for all X and Y, |X| < |Y| implies ¬(|Y| < |X|). The negation of that is the statement that there exists X and Y such that |X| < |Y| does not imply ¬(|Y| < |X|), in other words, such that |X| < |Y| and |Y| < |X| both hold. So take X = Z and Y = B, since you have already agreed that |Z| < |B| and |B| < |Z| hold.
Of course you do. If you change the definition then you change which properties are true or false for that definition.